
where the lower bound of 0 follows by assumption that \(\rho > 3/(4(a-1))\) . Assume without loss of generality that the unpenalized \(k\)  baseline parameters of \(\phi \)  and frailty log-variance parameter \(\sigma \)  are non-zero, so \(s \ge (k+1)\) . Then define \(S\)  to be the index set of the \(k+1\)  unpenalized elements, plus the \(s-k-1\)  largest elements of \(\beta \) . Then Lemma 5 of [1]} states that
\(3P_{\lambda }(\psi ^*) - P_{\lambda }({\widehat{\psi }}) \le {} 3\lambda \Vert \nu _{S}\Vert _1 - \lambda \Vert \nu _{S^{c}}\Vert _1.\) 
